We revisit the problem of computing shortest obstacle-avoiding paths among obstacles in three dimensions. We prove new hardness results, showing, e.g., that computing Euclidean shortest paths among sets of “stacked ” axis-aligned rectangles is NP-complete, and that computing L1-shortest paths among disjoint balls is NP-complete. On the positive side, we present an efficient algorithm for computing an L1shortest path between two given points that lies on or above a given polyhedral terrain. We also give polynomial-time algorithms for some versions of stacked polygonal obstacles that are “terrain-like ” and analyze the complexity of shortest path maps in the presence of parallel halfplane “walls.”

Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible convex hulls, measured by length and by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n log n) to O(n^13), and prove NP-hardness for some other variants.

How e#ciently can we search an unknown environment for a goal in unknown position? How much would it help if the environment were known? We answer these questions for simple polygons and for general graphs, by providing online search strategies that are as good as the best o#ine search algorithms, up to a constant factor. For other settings we prove that no such online algorithms exist.

Assume that an ordered set of imprecise points is given, where each point is specified by a region in which the point may lie. This set determines an imprecise polygon. We show that it is NP-complete to decide whether it is possible to place the points inside their regions in such a way that the resulting polygon is simple. Furthermore, it is NP-hard to minimize the length of a simple tour visiting the regions in order, when the connections between consecutive regions do not need to be straight line segments. 1

This paper studies the problem of finding the shortest path between two points in presence of single-point visibility constraints. In this type of constraints, there should be at least one point on the output path from which a fixed viewpoint is visible. The problem is studied in various domains including simple polygons, polygonal domains, polyhedral surfaces. The method is based on partitioning the boundary of the visibility region to a number of intervals according to shortest path structure of their points to both source and destination. The result for the two dimensional domains is worst-case optimal

So far, the best result in running time for solving the fixed watchman route problem (i.e., shortest path for viewing any point in a simple polygon with given start point) is O(n 3 log n), published in 2003 by M. Dror, A. Efrat, A. Lubiw, and J. Mitchell. – This paper provides an algorithm with κ(ε) · O(kn) runtime, where n is the number of vertices of the given simple polygon Π, and k the number of essential cuts; κ(ε) defines the numerical accuracy in dependency of a selected constant ε> 0. Moreover, our algorithm is significantly simpler, easier to understand and implement than previous ones for solving the fixed watchman route problem. 1

Abstract. Many surgical procedures could benefit from guiding a bevel-tip needle along circular arcs to multiple treatment points in a patient. At each treatment point, the needle can inject a radioactive pellet into a cancerous region or extract a tissue sample. Our main result is an algorithm to steer a bevel-tip needle through a sequence of treatment points in the plane while minimizing the number of times that the needle must be reoriented. This algorithm is related to [6] and takes quadratic time when consecutive points in the sequence are sufficiently separated. We can also guide a needle through an arbitrary sequence of points in the plane by accounting for a lack of optimal substructure.

Abstract Given a sequence s1,..., sK of K disjoint segments in the plane, a start point s and a target point t, we seek a path, that starts at s, visits in order each of the segments, and ends at t, such that the L1 length of the path is minimized. We give an O(K 2) algorithm that builds a data structure of size O(K) such that the shortest path, visiting k first segments in the sequence, to any point in the plane can be output in O(k) time. Related work In [2] Dror et al. solved the problem in the Euclidean metric. The touring problem can be formulated as a convex optimization program — [3] uses conic programming to find optimal tours in R d. Definitions and notation We say that a path π visits the sequence s1,..., sK if it starts at s and there exist points p1 ∈ s1,..., pK ∈ sK such that p1,..., pK appear in order along π. Let pk denote the first point of π (i.e., the point closest to s along

We consider a set of axis-parallel nonintersecting strips in the plane. An observer starts to the left of all strips and ends to the right, thus visiting all strips in the given order. A strip is inspected as long as the observer is inside the strip. How should the observer move to inspect the set of strips?

Many surgical procedures could benefit from guiding a bevel-tip needle through a sequence of treatment points in a patient. For example, brachytherapy procedures implant radioactive seeds to treat cancer, and biopsy